8.1 Polynomial Functions
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1 8.1 Polynomial Functions Algebra Goal 1: Evaluate polynomial functions. Goal : Identify general shapes of the graphs of polynomial functions. 1. What is a polynomial in one variable? Example 1: Determine if each expression is a polynomial in one variable. If so, determine its degree. If not, state the reason. a. x xy y b. a a 4 1 c. 1 n n d. 4 1b 6b 8 e. t 15t 9 f c 15c 4 6. Draw a graph of a polynomial function of degree zero to degree five.
2 Example : Determine whether each graph is odd or even. Then state how many real zeros each function has. Example 3: Find pm if 3. Example 4: Find f(3) f( 3) p( x) 3x 8x x if f ( x) x 4x 6. Example 5: Find f ( x h) for each function for the function f ( x) 3x 1. Example 6: Find f ( x h) for each function for the function f ( x) x x. Example 7: Find f ( x h) for each function for the function 3 f ( x) x x.
3 8. The Remainder and Factor Theorem Algebra Goal: Find factors of polynomials by using the factor theorem and synthetic division. Task 1: Let 1. Find f (). 4 3 f ( x) 3x x x 6. Use f( x) to perform the following tasks.. Use synthetic division to divide terms.) 4 3 f ( x) 3x x x 6 by x. (Don t forget to put zeroes in for missing 3. Compare the answer to number one and the remainder to number two. What did you notice? Task : Let 4 3 f ( x) x x 10x 5x 7. Use f( x) to perform the following tasks. (You will get a larger number than you are used to.) 4. Find f (8). 5. Use synthetic division to divide 4 3 f ( x) x x 10x 5x 7 by x Compare the answer to number four and the remainder to number five. What did you notice? Using synthetic division to find the value of a function is called. The Remainder Theorem: If a polynomial f( x) is divided by x a, the remainder is the constant f( a ), and Dividend = quotient divisor + remainder where qxis ( ) a polynomial with degree than the degree of f( x ).
4 Task 3: Let 7. Find f ( 4). 3 f ( x) x 4x x 4. Use f( x) to perform the following tasks. 8. Use synthetic division to divide 3 f ( x) x 4x x 4 by x Compare the answer to number seven and the remainder to number eight. What did you notice? 10. This remainder is different when compared to tasks 1 and. What do you think this means? The Factor Theorem: The binomial x ais a factor of the polynomial f( x) if and only if f( a). Example: Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. A ; x 5 B. 3 x x x 7 40 ; x 3 x x x
5 8.3 Graphing Polynomial Functions & Approximating Zeros Algebra Goal 1: To approximate the zeros of a polynomial function. Task 1: Graph the functions 4 f ( x) x 3x x 1 with the WINDOW below to get the graph show below. 1. Predict: Predict where you think the two zeros occur. Round your answers to one or two decimal places.. Discover: Use the table function to approximate the zeros of the function. A. Press nd followed by TBLSET and change where the table starts (TblStart) and the increment between the values of x in the table ( Tbl) to match the screen below. B. Now press nd followed by TABLE to view the table like the one above at the right. C. Now scroll up to the x value where you think the zeros occurs. D. Question 1: Between what two values of x does the zero occur? E. Question : How do you know? F. Question 3: After looking at the table, what is the approximate value of the zero? G. Now scroll down to the x value where you think the zeros occurs. H. Question 4: Between what two values of x does the zero occur? I. Question 5: How do you know? J. Question 6: After looking at the table, what is the approximate value of the zero?
6 Task 1: 3. Conclusion 1: Was your prediction correct? 4. Conclusion : Where do the zeros of a polynomial occur? 3 Task : Repeat the process above with the function g( x) x 5x 3x with the WINDOW below. 1. Sketch the graph on the grid provided below (Note: each column and row of dots represents one unit.).. Predict: Predict where you think the three zeros occur. Round your answers to one or two decimal places. 3. Discover: Use the table function to approximate the zeros of the function. A. Press nd followed by TBLSET and change where the table starts (TblStart) and the increment between the values of x in the table ( Tbl) to match the screen on page one. B. Now press nd followed by TABLE to view the table. C. Now scroll to the values of x where you think the zeros occurs. D. Question 1: 1 st zero: Between what two values of x does the zero occur? E. Question : What is the approximate value of the 1 st zero? F. Question 3: nd zero: Between what two values of x does the zero occur? G. Question 4: What is the approximate value of the nd zero? H. Question 5: 3 rd zero: Between what two values of x does the zero occur? I. Question 6: What is the approximate value of the 3 rd zero? Task : 4. Conclusion 1: Was your prediction correct? 5. Conclusion : Where do the zeros of a polynomial occur?
7 8.4 Roots and Zeros Algebra Fundamental Theorem of Algebra: 3 Example 1: Find all roots of 0 x x 4x 4. 3 Example : Find all roots of x 3x 10x 4 0 Complex Conjugates Theorem: Example 3: Find all zeros of 3 g( x) x 4x 6x 4 if is one zero of f( x ).
8 Example 4: Find all zeros of 3 f ( x) x 5x 7x 51if 4 i is one zero of f( x ). Example 5: Find all zeros of 3 f ( x) x 7x 16x 10 if 3i is one zero of f( x ). Descartes Rule of Signs: Go back to each example and perform this rule.
9 8.6 Quadratic Techniques Algebra Goal: Solve nonquadratic equations by using quadratic techniques. Example 1: Solve 4 3 x 3x 18x 0 Example : Solve t Quadratic Form: a f x b f x c 0 Write the following equations in quadratic form, if possible. A. x 4x 4 0 B x 7x 8 0 C. 4 y 10y 8 0 D. x x 0 0 E. x x 3 0 F. z3 z 18 0 Example 3: Solve x 7x 1 0 Example 4: Solve 4 4 x 17x 16 0
10 Example 5: x x 1 0 Example 6: k k 10 0 Example 7: Solve x x 15 0 Example 8: Solve y y 36 0 Example 9: Solve y8 y 7 0 Example 10: p10 p 4 0
11 Goal: Find the composition of functions. 8.7 Composition of Functions Algebra Pre-Activity: Put the following situations in the proper order. Food chain: Complete the given food chain by putting the following animals in order according to their place in the food chain. a snake, a fly, and a frog Sending a text: What is the proper order for sending a text? open texting app, sending message, typing message Peanut Butter Sandwich: How do you make a peanut butter sandwich? spread peanut butter on bread, put pieces of bread together, take bread out of the bag What must happen for all of the above situations to be successful? Put the above situations in function notation. Mark the first step with the letter x, the second with the letter g, and the final step with the letter f. Composition of functions: Just like the situations above, a composition of functions must be done in a specific order, where the each step is dependent on the one before it. Notation:
12 Example 1: If f ( x) x and g( x) x, Example : If f ( x) x and g( x) x, find f g 3. find g f 3. Example 3: If f ( x) x 7 and g x ( ) x 4, Example 4: If f ( x) x 7 and find f g. find g f. g x ( ) x 4, Example 5: If f x find ( ) x 4 and g( x) 4x 1, Example 6: If f gx. find f ( x) x x 3and g( x) 4x, g f x. Example 7: If find f ( x) x 3x 4 and g( x) x 3, Example 8: If f x 3 ( ) x 3 f g x. find and g( x) x 7, g f x. Example 9: Find g h 4 and h g 5 Example 10: Find g h and h g 6 g 1,, 3,10, 5,6 g 6,, 6, 7, 9,0 h 6,9, 4,1, 3, 1 h,0,,9, 3,5
13 8.8 Inverse Functions and Relations Algebra Goal 1: Determine the inverse of a function or relation. Goal : Graph functions and their inverses. Goal 3: Work backward to solve problems. Definition of Inverse Functions: Graphs of Inverse Functions: Example 1: Determine whether f ( x) 6 xand f gx g f x 6 x gx ( ) are inverse functions. Example : Determine whether f ( x) 3x 4 and f gx g f x x 4 gx ( ) are inverse functions. 3 Example 3: Find the inverse of each relation and determine whether the inverse is a function. A. f 0,3, 4,5,,1, 5,9 B. f,4, 3,7, 0,4, 7, 1 f 1 f Function? Why or why not? Function? Why or why not?
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